What is a Sufficient Statistic 🌱
Definition
If Y1,…,Yn are iid samples drawn from a distribution with parameter θ then a statistic U=U(Y1,…,Yn) is said to be sufficient for θ iff the conditional distribution of the data Y1,…,Yn∣U does not depend on the parameter θ.
Method to Find Them
If the likelihood can be expressed as the following, then U(Y1,…,Yn) is a sufficient statistic
L(θ;y1,y2,…,yn)=g(U(y1,y2,…,yn),θ)h(y1,y2,…,yn)Minimum Sufficient Statistic
A minimal sufficient statistic T(Y1,…,Yn) is a sufficient statistic that can be expressed as some function of any other sufficient statistic ˜T(Y1,…,Yn):
T(y1,y2,…,yn)=g(˜T(y1,y2,…,yn))Finding a minimal sufficient statistic
Let x1,…,xn∼f(x;θ) and y1,…,yn∼f(x;θ) be two samples from the same distribution and T be a statistic. Suppose:
R=R(x1,…,xn,y1,…,yn,θ)=f(x1,…,xn;θ)f(y1,…,yn;θ)=L(θ;x1,…,xn)L(θ;y1,…,yn)If for all θ where R is defined we have:
R does not depend on θ⟺T(x1,…,xn)=T(y1,…,yn)then T is a minimal sufficient statistic.
Rao - Blackwell Theorem
Let ˆθ be an unbiased estimator for θ such that V(ˆθ)<∞. If U is a sufficient statistic for θ, then for all θ:
E(E(ˆθ∣U))=θ and V(E(ˆθ∣U))≤V(ˆθ)Notes mentioning this note
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